Question

If a and b are rational numbers and 5+2root3 upon 7+4root3 = a-b root 3, find the value of a and b.​

Answer 1

Question :-

If a and b are rational numbers and $\frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = a - b \sqrt{3}$ , find the values of a and b.

Required Solution :-

$\frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = \frac{5 + 2\sqrt{3} }{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } </p><p>= \frac{35 - 20 \sqrt{3 } + 14 \sqrt{3} - 8 \times 3}{ {7}^{2} - (4 \sqrt{3}) ^{2} }$

$\frac{11 - 6 \sqrt{3} }{49 - 48} = \frac{11 - 6 \sqrt{3} }{1} = 11 - 6 \sqrt{3}$

∴ a - b$\sqrt{3}$ = 11 - 6 $\sqrt{3}$

⟹ A = 11 and B = 6